<!DOCTYPE html>
<html lang="zh-CN">
<head>
  <meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1, maximum-scale=2">
<meta name="theme-color" content="#222">
<meta name="generator" content="Hexo 4.2.1">
  <link rel="icon" type="image/png" sizes="32x32" href="/images/32x32.png">
  <link rel="icon" type="image/png" sizes="16x16" href="/images/16x16.png">

<link rel="stylesheet" href="/css/main.css">

<link rel="stylesheet" href="//fonts.googleapis.com/css?family=Lato:300,300italic,400,400italic,700,700italic|Lato', 'Microsoft Yahei Light:300,300italic,400,400italic,700,700italic|Cambria', 'Microsoft Yahei Light:300,300italic,400,400italic,700,700italic|Verdana', Lato, 'Microsoft Yahei Light:300,300italic,400,400italic,700,700italic&display=swap&subset=latin,latin-ext">
<link rel="stylesheet" href="/lib/font-awesome/css/all.min.css">

<script id="hexo-configurations">
    var NexT = window.NexT || {};
    var CONFIG = {"hostname":"yoursite.com","root":"/","scheme":"Mist","version":"7.8.0","exturl":false,"sidebar":{"position":"right","display":"hide","padding":18,"offset":12,"onmobile":false},"copycode":{"enable":false,"show_result":false,"style":null},"back2top":{"enable":true,"sidebar":false,"scrollpercent":false},"bookmark":{"enable":false,"color":"#222","save":"auto"},"fancybox":false,"mediumzoom":false,"lazyload":false,"pangu":false,"comments":{"style":"tabs","active":null,"storage":true,"lazyload":false,"nav":null},"algolia":{"hits":{"per_page":10},"labels":{"input_placeholder":"Search for Posts","hits_empty":"We didn't find any results for the search: ${query}","hits_stats":"${hits} results found in ${time} ms"}},"localsearch":{"enable":true,"trigger":"auto","top_n_per_article":1,"unescape":false,"preload":false},"motion":{"enable":true,"async":false,"transition":{"post_block":"fadeIn","post_header":"slideDownIn","post_body":"slideDownIn","coll_header":"slideLeftIn","sidebar":"slideUpIn"}},"path":"search.xml"};
  </script>

  <meta name="description" content="Problem 690 Tom and JerryTom (the cat) and Jerry (the mouse) are playing on a simple graph $G$. Every vertex of $G$ is a mousehole, and every edge of $G$ is a tunnel connecting two mouseholes. Origin">
<meta property="og:type" content="article">
<meta property="og:title" content="Problem 690">
<meta property="og:url" content="http://yoursite.com/690/index.html">
<meta property="og:site_name" content="Project Euler | 欧拉计划">
<meta property="og:description" content="Problem 690 Tom and JerryTom (the cat) and Jerry (the mouse) are playing on a simple graph $G$. Every vertex of $G$ is a mousehole, and every edge of $G$ is a tunnel connecting two mouseholes. Origin">
<meta property="og:locale" content="zh_CN">
<meta property="og:image" content="https://projecteuler.net/project/images/p690_graphs.jpg">
<meta property="og:image" content="https://projecteuler.net/project/images/p690_graphs.jpg">
<meta property="article:published_time" content="2019-11-24T15:00:00.000Z">
<meta property="article:modified_time" content="2020-11-06T22:28:29.200Z">
<meta property="article:author" content="sx349">
<meta name="twitter:card" content="summary">
<meta name="twitter:image" content="https://projecteuler.net/project/images/p690_graphs.jpg">

<link rel="canonical" href="http://yoursite.com/690/">


<script id="page-configurations">
  // https://hexo.io/docs/variables.html
  CONFIG.page = {
    sidebar: "",
    isHome : false,
    isPost : true,
    lang   : 'zh-CN'
  };
</script>

  <title>Problem 690 | Project Euler | 欧拉计划</title>
  






  <noscript>
  <style>
  .use-motion .brand,
  .use-motion .menu-item,
  .sidebar-inner,
  .use-motion .post-block,
  .use-motion .pagination,
  .use-motion .comments,
  .use-motion .post-header,
  .use-motion .post-body,
  .use-motion .collection-header { opacity: initial; }

  .use-motion .site-title,
  .use-motion .site-subtitle {
    opacity: initial;
    top: initial;
  }

  .use-motion .logo-line-before i { left: initial; }
  .use-motion .logo-line-after i { right: initial; }
  </style>
</noscript>

</head>

<body itemscope itemtype="http://schema.org/WebPage">
  <div class="container use-motion">
    <div class="headband"></div>

    <header class="header" itemscope itemtype="http://schema.org/WPHeader">
      <div class="header-inner"><div class="site-brand-container">
  <div class="site-nav-toggle">
    <div class="toggle" aria-label="切换导航栏">
      <span class="toggle-line toggle-line-first"></span>
      <span class="toggle-line toggle-line-middle"></span>
      <span class="toggle-line toggle-line-last"></span>
    </div>
  </div>

  <div class="site-meta">

    <a href="/" class="brand" rel="start">
      <span class="logo-line-before"><i></i></span>
      <h1 class="site-title">Project Euler | 欧拉计划</h1>
      <span class="logo-line-after"><i></i></span>
    </a>
  </div>

  <div class="site-nav-right">
    <div class="toggle popup-trigger">
        <i class="fa fa-search fa-fw fa-lg"></i>
    </div>
  </div>
</div>




<nav class="site-nav">
  <ul id="menu" class="main-menu menu">
        <li class="menu-item menu-item-home">

    <a href="/" rel="section"><i class="fa fa-home fa-fw"></i>首页</a>

  </li>
        <li class="menu-item menu-item-problem">

    <a href="/problems" rel="section"><i class="fa fa-tags fa-fw"></i>题目</a>

  </li>
        <li class="menu-item menu-item-about">

    <a href="/about/" rel="section"><i class="fa fa-user fa-fw"></i>关于</a>

  </li>
      <li class="menu-item menu-item-search">
        <a role="button" class="popup-trigger"><i class="fa fa-search fa-fw"></i>搜索
        </a>
      </li>
  </ul>
</nav>



  <div class="search-pop-overlay">
    <div class="popup search-popup">
        <div class="search-header">
  <span class="search-icon">
    <i class="fa fa-search"></i>
  </span>
  <div class="search-input-container">
    <input autocomplete="off" autocapitalize="off"
           placeholder="搜索..." spellcheck="false"
           type="search" class="search-input">
  </div>
  <span class="popup-btn-close">
    <i class="fa fa-times-circle"></i>
  </span>
</div>
<div id="search-result">
  <div id="no-result">
    <i class="fa fa-spinner fa-pulse fa-5x fa-fw"></i>
  </div>
</div>

    </div>
  </div>

</div>
    </header>

    
  <div class="back-to-top">
    <i class="fa fa-arrow-up"></i>
    <span>0%</span>
  </div>


    <main class="main">
      <div class="main-inner">
        <div class="content-wrap">
          

          <div class="content post posts-expand">
            

    
  
  
  <article itemscope itemtype="http://schema.org/Article" class="post-block" lang="zh-CN">
    <link itemprop="mainEntityOfPage" href="http://yoursite.com/690/">

    <span hidden itemprop="author" itemscope itemtype="http://schema.org/Person">
      <meta itemprop="image" content="/images/avatar.png">
      <meta itemprop="name" content="sx349">
      <meta itemprop="description" content="Project Euler | 欧拉计划 中文翻译站">
    </span>

    <span hidden itemprop="publisher" itemscope itemtype="http://schema.org/Organization">
      <meta itemprop="name" content="Project Euler | 欧拉计划">
    </span>
      <header class="post-header">
        <h1 class="post-title" itemprop="name headline">
          Problem 690
        </h1>

        <div class="post-meta">
            <span class="post-meta-item">
              <span class="post-meta-item-icon">
                <i class="far fa-calendar"></i>
              </span>
              <span class="post-meta-item-text">题目发布于</span>

              <time title="发布时间：2019-11-24 07:00:00" itemprop="dateCreated datePublished" datetime="2019-11-24T07:00:00-08:00">2019-11-24</time>
            </span>
              <span class="post-meta-item">
                <span class="post-meta-item-icon">
                  <i class="far fa-calendar-check"></i>
                </span>
                <span class="post-meta-item-text">翻译更新于</span>
                <time title="更新时间：2020-11-06 14:28:29" itemprop="dateModified" datetime="2020-11-06T14:28:29-08:00">2020-11-06</time>
              </span>

          

        </div>
      </header>

    
    
    
    <div class="post-body" itemprop="articleBody">

      
        <hr>
<h1 id="Problem-690"><a href="#Problem-690" class="headerlink" title="Problem 690"></a><a href="https://projecteuler.net/problem=690" target="_blank" rel="noopener">Problem 690</a></h1><hr>
<h2 id="Tom-and-Jerry"><a href="#Tom-and-Jerry" class="headerlink" title="Tom and Jerry"></a><strong>Tom and Jerry</strong></h2><p>Tom (the cat) and Jerry (the mouse) are playing on a simple graph $G$.</p>
<p>Every vertex of $G$ is a mousehole, and every edge of $G$ is a tunnel connecting two mouseholes.</p>
<p>Originally, Jerry is hiding in one of the mouseholes.<br>Every morning, Tom can check one (and only one) of the mouseholes. If Jerry happens to be hiding there then Tom catches Jerry and the game is over.<br>Every evening, if the game continues, Jerry moves to a mousehole which is adjacent (i.e. connected by a tunnel, if there is one available) to his current hiding place. The next morning Tom checks again and the game continues like this.</p>
<p>Let us call a graph $G$ a <i>Tom graph</i>, if our super-smart Tom, who knows the configuration of the graph but does not know the location of Jerry, can <i>guarantee</i> to catch Jerry in finitely many days. For example consider all graphs on $3$ nodes:</p>
<p><img src="https://projecteuler.net/project/images/p690_graphs.jpg" alt="&quot;Graphs on 3 nodes&quot;"></p>
<p>For graphs $1$ and $2$, Tom will catch Jerry in at most three days. For graph $3$ Tom can check the middle connection on two consecutive days and hence guarantee to catch Jerry in at most two days. These three graphs are therefore Tom Graphs. However, graph $4$ is not a Tom Graph because the game could potentially continue forever.</p>
<p>Let $T(n)$ be the number of different Tom graphs with $n$ vertices. Two graphs are considered the same if there is a bijection $f$ between their vertices, such that $(v,w)$ is an edge if and only if $(f(v),f(w))$ is an edge.</p>
<p>We have $T(3) = 3$, $T(7) = 37$, $T(10) = 328$ and $T(20) = 1416269$.</p>
<p>Find $T(2019)$ giving your answer modulo $1\ 000\ 000\ 007$.</p>
<hr>
<h2 id="猫和老鼠"><a href="#猫和老鼠" class="headerlink" title="猫和老鼠"></a><strong>猫和老鼠</strong></h2><p>汤姆（猫）和杰瑞（老鼠）在玩游戏，他们玩游戏的地图可以用简单图$G$表示。</p>
<p>$G$中的每一个顶点代表一个老鼠洞，而$G$中的每一条边代表连通两个老鼠洞的一条地道。</p>
<p>一开始，杰瑞躲在其中一个老鼠洞中。<br>每天早上，汤姆可以查看一个老鼠洞。如果杰瑞恰好躲在这个老鼠洞中，那么汤姆抓住杰瑞，游戏结束。<br>每天晚上，如果游戏没有结束，杰瑞会从当前藏身的老鼠洞移动到一个相邻（有地道直接相连）的老鼠洞。第二天早上再次轮到汤姆查看一个老鼠洞，依此类推。</p>
<p>汤姆不知道杰瑞的位置，但他知道图$G$的形状，如果超级聪明的汤姆能<i class=zh>保证</i>在有限天内抓住杰瑞，那么我们称图$G$为<i class=zh>汤姆图</i>。考虑如下有$3$个顶点的图：</p>
<p><img src="https://projecteuler.net/project/images/p690_graphs.jpg" alt="“有3个顶点的图”"></p>
<p>对于图$1$和$2$，汤姆可以在至多三天内抓住杰瑞。对于图$3$，汤姆只需连续两天都检查右下角的老鼠洞，就能保证在至多两天内抓住杰瑞。因此这三张图都是汤姆图。然而，图$4$不是汤姆图，因为这个游戏有可能永远不会结束。</p>
<p>记$T(n)$为有$n$个顶点的不同汤姆图的数量。如果两张图的顶点间存在双射$f$，使得$(v,w)$是第一张图的一条边当且仅当$(f(v),f(w))$是第二张图的一条边，这两张图就被认为是相同的图。</p>
<p>已知$T(3) = 3$，$T(7) = 37$，$T(10) = 328$，$T(20) = 1416269$。</p>
<p>求$T(2019)$并将你的答案对$1\ 000\ 000\ 007$取余。</p>
<hr>

    </div>

    
    
    

      <footer class="post-footer">

        


        
    <div class="post-nav">
      <div class="post-nav-item">
    <a href="/689/" rel="prev" title="Problem 689">
      <i class="fa fa-chevron-left"></i> Problem 689
    </a></div>
      <div class="post-nav-item">
    <a href="/691/" rel="next" title="Problem 691">
      Problem 691 <i class="fa fa-chevron-right"></i>
    </a></div>
    </div>
      </footer>
    
  </article>
  
  
  



          </div>
          
    <div class="comments" id="gitalk-container"></div>

<script>
  window.addEventListener('tabs:register', () => {
    let { activeClass } = CONFIG.comments;
    if (CONFIG.comments.storage) {
      activeClass = localStorage.getItem('comments_active') || activeClass;
    }
    if (activeClass) {
      let activeTab = document.querySelector(`a[href="#comment-${activeClass}"]`);
      if (activeTab) {
        activeTab.click();
      }
    }
  });
  if (CONFIG.comments.storage) {
    window.addEventListener('tabs:click', event => {
      if (!event.target.matches('.tabs-comment .tab-content .tab-pane')) return;
      let commentClass = event.target.classList[1];
      localStorage.setItem('comments_active', commentClass);
    });
  }
</script>

        </div>
          
  
  <div class="toggle sidebar-toggle">
    <span class="toggle-line toggle-line-first"></span>
    <span class="toggle-line toggle-line-middle"></span>
    <span class="toggle-line toggle-line-last"></span>
  </div>

  <aside class="sidebar">
    <div class="sidebar-inner">

      <ul class="sidebar-nav motion-element">
        <li class="sidebar-nav-toc">
          文章目录
        </li>
        <li class="sidebar-nav-overview">
          站点概览
        </li>
      </ul>

      <!--noindex-->
      <div class="post-toc-wrap sidebar-panel">
          <div class="post-toc motion-element"><ol class="nav"><li class="nav-item nav-level-1"><a class="nav-link" href="#Problem-690"><span class="nav-text">Problem 690</span></a><ol class="nav-child"><li class="nav-item nav-level-2"><a class="nav-link" href="#Tom-and-Jerry"><span class="nav-text">Tom and Jerry</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="#猫和老鼠"><span class="nav-text">猫和老鼠</span></a></li></ol></li></ol></div>
      </div>
      <!--/noindex-->

      <div class="site-overview-wrap sidebar-panel">
        <div class="site-author motion-element" itemprop="author" itemscope itemtype="http://schema.org/Person">
    <img class="site-author-image" itemprop="image" alt="sx349"
      src="/images/avatar.png">
  <p class="site-author-name" itemprop="name">Leonhard Euler(1707-1783)</p>
  <p class="site-description motion-element">Project Euler | 欧拉计划<br>中文翻译站</p>
</div>
  <div class="cc-license motion-element" itemprop="license">
    <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/" class="cc-opacity" rel="noopener" target="_blank"><img src="/images/cc-by-nc-sa.svg" alt="Creative Commons"></a>
  </div>


  <div class="links-of-blogroll motion-element">
    <div class="links-of-blogroll-title"><i class="fa fa-link fa-fw"></i>
      友情链接
    </div>
    <ul class="links-of-blogroll-list">
        <li class="links-of-blogroll-title">
          <a href="https://projecteuler.net/" title="https:&#x2F;&#x2F;projecteuler.net" rel="noopener" target="_blank">Project Euler</a>
        </li>
        <li class="links-of-blogroll-title">
          <a href="http://jakwings.is-programmer.com/Project_Euler" title="http:&#x2F;&#x2F;jakwings.is-programmer.com&#x2F;Project_Euler" rel="noopener" target="_blank">饮水思源汉化</a>
        </li>
        <li class="links-of-blogroll-title">
          <a href="http://sx349.github.io/" title="http:&#x2F;&#x2F;sx349.github.io" rel="noopener" target="_blank">译者博客</a>
        </li>
    </ul>
  </div>

      </div>

    </div>
  </aside>
  <div id="sidebar-dimmer"></div>


      </div>
    </main>

    <footer class="footer">
      <div class="footer-inner">
        

        

<div class="copyright">
  
  &copy; 2001 – 
  <span itemprop="copyrightYear">2021</span>
  <span class="with-love">
    <i class="fas fa-calculator"></i>
  </span>
  <span class="author" itemprop="copyrightHolder">Project Euler | 欧拉计划</span>
</div>
  <div class="powered-by">由 <a href="https://hexo.io/" class="theme-link" rel="noopener" target="_blank">Hexo</a> & <a href="https://mist.theme-next.org/" class="theme-link" rel="noopener" target="_blank">NexT.Mist</a> 强力驱动
  </div>

        








      </div>
    </footer>
  </div>

  
  <script src="/lib/anime.min.js"></script>
  <script src="/lib/velocity/velocity.min.js"></script>
  <script src="/lib/velocity/velocity.ui.min.js"></script>

<script src="/js/utils.js"></script>

<script src="/js/motion.js"></script>


<script src="/js/schemes/muse.js"></script>


<script src="/js/next-boot.js"></script>




  




  
<script src="/js/local-search.js"></script>













  

  
      

<script>
  if (typeof MathJax === 'undefined') {
    window.MathJax = {
      loader: {
        source: {
          '[tex]/amsCd': '[tex]/amscd',
          '[tex]/AMScd': '[tex]/amscd'
        }
      },
      tex: {
        inlineMath: {'[+]': [['$', '$']]},
        tags: 'ams'
      },
      options: {
        renderActions: {
          findScript: [10, doc => {
            document.querySelectorAll('script[type^="math/tex"]').forEach(node => {
              const display = !!node.type.match(/; *mode=display/);
              const math = new doc.options.MathItem(node.textContent, doc.inputJax[0], display);
              const text = document.createTextNode('');
              node.parentNode.replaceChild(text, node);
              math.start = {node: text, delim: '', n: 0};
              math.end = {node: text, delim: '', n: 0};
              doc.math.push(math);
            });
          }, '', false],
          insertedScript: [200, () => {
            document.querySelectorAll('mjx-container').forEach(node => {
              let target = node.parentNode;
              if (target.nodeName.toLowerCase() === 'li') {
                target.parentNode.classList.add('has-jax');
              }
            });
          }, '', false]
        }
      }
    };
    (function () {
      var script = document.createElement('script');
      script.src = '//cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js';
      script.defer = true;
      document.head.appendChild(script);
    })();
  } else {
    MathJax.startup.document.state(0);
    MathJax.texReset();
    MathJax.typeset();
  }
</script>

    

  

<link rel="stylesheet" href="//cdn.jsdelivr.net/npm/gitalk@1/dist/gitalk.min.css">

<script>
NexT.utils.loadComments(document.querySelector('#gitalk-container'), () => {
  NexT.utils.getScript('//cdn.jsdelivr.net/npm/gitalk@1/dist/gitalk.min.js', () => {
    var gitalk = new Gitalk({
	  title       : document.title.substr(0,29),
      clientID    : '4ba99a8839c5e7dde683',
      clientSecret: '3984f7cad409553d6afde8dbc2c08fc425e7bbdc',
      repo        : 'pe-cn-comments',
      owner       : 'pe-cn',
      admin       : ['sx349'],
      id          : 'e6854d784bba5f0d0441a290c8a930b3',
        language: '',
      distractionFreeMode: true
    });
    gitalk.render('gitalk-container');
  }, window.Gitalk);
});
</script>

</body>
</html>
